Digital homotopy fixed point theory

Obstructions to Homotopy Invariance in Parametrized Fixed Point Theory

In “zero-parameter” or classical Nielsen fixed point theory one studies Fix(f) := {x ∈ X | f(x) = x} where f : X → X is a map. In case X is an oriented compact manifold and f is transverse to the identity map, idX , Fix(f) is a finite set each of whose elements carries a natural sign, ±1, the index of that fixed point. The set Fix(f) is partitioned into Nielsen classes. Adding the indices withi...

Relative Fixed Point Theory

The Lefschetz fixed point theorem and its converse have many generalizations. One of these generalizations is to endomorphisms of a space relative to a fixed subspace. In this paper we define relative Lefschetz numbers and Reidemeister traces using traces in bicategories with shadows. We use the functoriality of this trace to identify different forms of these invariants and to prove a relative ...

Epsilon Nielsen Fixed Point Theory

Let f : X → X be a map of a compact, connected Riemannian manifold, with or without boundary. For > 0 sufficiently small, we introduce an -Nielsen number N ( f ) that is a lower bound for the number of fixed points of all self-maps of X that are -homotopic to f . We prove that there is always a map g : X → X that is -homotopic to f such that g has exactlyN ( f ) fixed points. We describe proced...

The Rational Homotopy of Fixed Point Sets of Torus Actions

The Lubin-tate Spectrum and Its Homotopy Fixed Point Spectra

This note is a summary of the results of my Ph.D. thesis (plus slight modifications), completed May 9, 2003, under the supervision of Professor Paul Goerss at Northwestern University. Let En be the Lubin-Tate spectrum with En∗ = W (Fpn)[[u1, ..., un−1]][u], where the degree of u is −2 and the complete power series ring over the Witt vectors is in degree zero. Let Gn = Sn o Gal(Fpn/Fp), where Sn...